Conservation of population size is required for self-organized criticality in evolution models Y. Murase1, P. A. Rikvold2,3 1RIKEN Center for Computational Science 2Florida State Univ., 3Univ. of Oslo ref. Y.Murase and P.A. Rikvold, New J. Phys. 20, 083023 (2018)

The Bak-Sneppen model • a minimal model for biological coevolution • shows SOC • critical avalanche of extinctions • intermittent dynamics • assumptions: • Darwinian competition • interspecies interactions Bak-Sneppen, PRL (1983)

fitness (a scalar value [0,1]) loop { i = argmin( f ) t += t_0 * exp( f[i] / f_0 ) f[i] = rand(0,1) f[i+1] = rand(0,1) f[i-1] = rand(0,1) } extinction of species i interspecies interaction Arrhenius' type function ⌧ext(fi) / exp (fi/f0) (null) (null) (null) (null)

spontaneous emergence of the threshold Bak-Sneppen, PRL (1993)

spontaneous emergence of the threshold intermittent evolutionary dynamics (punctuated equilibria) Bak-Sneppen, PRL (1993)

spontaneous emergence of the threshold intermittent evolutionary dynamics (punctuated equilibria) power-law avalanche size (mass extinctions) Bak-Sneppen, PRL (1993)

Population dynamics models • population dynamics + species introduction • e.g. dynamical system with variable system size loop { species.each {|i| x[i] = update_pop(x) } species.each {|i| extinction(i) if x[i] <= 0 } add_new_species() } e.g., Tangled-Nature model, scale invariant model, replicator equations, web world model, ... ref. H.J.Jensen, Eur. J. Phys. (2018) new species extinction

Population dynamics models with migration rule • Migration rule : interspecies interactions are randomly determined irrespective of the existing species • c.f. Mutation rule : new species are made based on the existing species Y.Murase et al., J. Theor. Biol. (2010) • 1/f2 fluctuations • exp. extinction sizes • skewed lifetime distribution non-SOC

Dynamical graph model • A minimal model for the class of migration models. • Population dynamics is replaced by a simple graph dynamics. • Species can survive as long as its incoming link weight ≥ 0. Y.Murase et al., NJP (2010) from T.Shimada, Sci.Rep (2014)

• belongs to the same class as migration pop. dynamics models • 1/f2 fluctuations • exp. extinction sizes • skewed lifetime distribution "Ising" model for population dynamics models

SOC non-SOC BS model DG model population dynamics models Darwinian competition & successive introduction of new species common assumptions

What is the key factor to yield SOC/non-SOC behaviors?

Key differences b/w BS & DG

Key differences b/w BS & DG 1. number of species (N) • BS : N is fixed to a value given as a model parameter. • DG : N changes according to evolutionary process.

Key differences b/w BS & DG 1. number of species (N) • BS : N is fixed to a value given as a model parameter. • DG : N changes according to evolutionary process. 2. the extinction threshold (fth ) • BS : fth is self-organized as a result of evolutionary process. • DG : fth is predefined

Key differences b/w BS & DG 1. number of species (N) • BS : N is fixed to a value given as a model parameter. • DG : N changes according to evolutionary process. 2. the extinction threshold (fth ) • BS : fth is self-organized as a result of evolutionary process. • DG : fth is predefined 3. instant / regular migrations • BS : f-dependent τext followed by instant migration τimg =0 • DG : regular migration τimg =1 & f-dependent τext

Key differences b/w BS & DG 1. number of species (N) • BS : N is fixed to a value given as a model parameter. • DG : N changes according to evolutionary process. 2. the extinction threshold (fth ) • BS : fth is self-organized as a result of evolutionary process. • DG : fth is predefined 3. instant / regular migrations • BS : f-dependent τext followed by instant migration τimg =0 • DG : regular migration τimg =1 & f-dependent τext 4. fitness definition (f) • BS : node-based • DG : link-based, i.e., fi = ∑wji

Model 1: link-based BS model • fixed N (as in BS model) • eliminate minimum fitness species followed by an immediate introduction of new species (as in BS model) • increment time by τext ∝ exp(fmin /f0 ), • represented by a weighted directed network (as in DG model) • fi = ∑ wji • new species has new links drawn randomly (as in DG model)

0 0.4 0.8 1.2 1.6 0 0.2 0.4 0.6 0.8 1 P(f) fitness, f 10-6 10-5 10-4 10-3 10-2 10-1 100 0 5 10 15 20 25 30 35 exp(-s/s0 ) P(s) extinction size, s 0 0.04 0.08 0.12 0.16 0.2 -4 0 4 8 12 P(f) fitness, f 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 100 101 102 103 104 exp(-(L/L0 )0.6) P(L) species lifetime, L 10-5 10-4 10-3 10-2 10-1 100 0 2 4 6 8 10 exp(-τ/τ0 ) P(τ) inter-event time, τ 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 100 101 102 103 s-1.5 P(s) extinction size, s 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 106 108 1010 L-1 exp(-L/L0 ) P(L) species lifetime, L 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 100 101 102 103 104 105 106 107 108 τ-1 P(τ) inter-event time, τ 0 0.04 0.08 0.12 0.16 0.2 -4 0 4 8 12 P(f) fitness, f 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 100 101 102 103 104 s-1.5 P(s) extinction size, s 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 10-2 100 102 104 106 108 1010 1012 L-1 P(L) species lifetime, L 10-15 10-10 10-5 100 10-2 100 102 104 106 108 1010 1012 τ-1.2 P(τ) inter-event time, τ BS model DG model link-based BS model (a-1) (a-2) (a-3) (a-4) (b-1) (b-2) (b-3) (b-4) (c-1) (c-2) (c-3) (c-4) mean-field

0 0.4 0.8 1.2 1.6 0 0.2 0.4 0.6 0.8 1 P(f) fitness, f 10-6 10-5 10-4 10-3 10-2 10-1 100 0 5 10 15 20 25 30 35 exp(-s/s0 ) P(s) extinction size, s 0 0.04 0.08 0.12 0.16 0.2 -4 0 4 8 12 P(f) fitness, f 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 100 101 102 103 104 exp(-(L/L0 )0.6) P(L) species lifetime, L 10-5 10-4 10-3 10-2 10-1 100 0 2 4 6 8 10 exp(-τ/τ0 ) P(τ) inter-event time, τ 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 100 101 102 103 s-1.5 P(s) extinction size, s 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 106 108 1010 L-1 exp(-L/L0 ) P(L) species lifetime, L 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 100 101 102 103 104 105 106 107 108 τ-1 P(τ) inter-event time, τ 0 0.04 0.08 0.12 0.16 0.2 -4 0 4 8 12 P(f) fitness, f 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 100 101 102 103 104 s-1.5 P(s) extinction size, s 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 10-2 100 102 104 106 108 1010 1012 L-1 P(L) species lifetime, L 10-15 10-10 10-5 100 10-2 100 102 104 106 108 1010 1012 τ-1.2 P(τ) inter-event time, τ BS model DG model link-based BS model (a-1) (a-2) (a-3) (a-4) (b-1) (b-2) (b-3) (b-4) (c-1) (c-2) (c-3) (c-4) mean-field

SOC non-SOC BS model DG model link-based BS model population dynamics models node-based / link-based is not the key factor

Model 2: generalized model 1. fixed / variable N 2. self-organized / fixed fth 3. instant / regular immigrations 4. node-based / link-based analogous to canonical / grand-canonical ensembles (fixed N vs fixed μ) DG model link-based BS model

Model 2: generalized model 1. fixed / variable N 2. self-organized / fixed fth 3. instant / regular immigrations 4. node-based / link-based analogous to canonical / grand-canonical ensembles (fixed N vs fixed μ) generalized model DG model link-based BS model

model definition loop { i = argmin( f ) if tau_ext(f[i]) < tau_mig(N) extinction(i) t += tau_ext(f[i]) else add_new_species t += tau_mig(N) end } ⌧ext(f) = exp (f/f0) (null) (null) (null) (null) ⌧mig(N) = exp (µ(N N0)/f0) (null) (null) (null) (null) μ : parameter to control the fluctuation of N around N0 ( N > N0 ) ⌧mig %, N & N < N0 ) ⌧mig &, N % (null) (null) (null) (null)

generalized model DG model link-based BS model µ = 0 (null) (null) (null) (null) µ ! 1 (null) (null) (null) (null) ⌧ext(f) = exp (f/f0) (null) (null) (null) (null) ⌧mig(N) = exp (µ(N N0)/f0) (null) (null) (null) (null) t extinction migration t

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µ = 0 (null) (null) (null) (null) µ = 0.1 (null) (null) (null) (null)

10-10 10-8 10-6 10-4 10-2 100 100 101 102 103 104 s-1.5 P(s) extinction size, s µ = 0 10-4 10-3 10-2 10-1 0 0.1 0.2 -10 -5 0 5 10 P(f) fitness, f µ = 0 10-4 10-3 10-2 10-1 0 0.1 -0.3 0 0.3 10-16 10-12 10-8 10-4 100 10-2 100 102 104 106 108 1010 1012 L-1 exp(-(L/L0 )1/2) P(L) species lifetime, L µ = 0 10-4 10-3 10-2 10-1 10-20 10-15 10-10 10-5 100 10-2 100 102 104 106 108 1010 1012 τ-1.2 P(τ) inter-event time, τ µ = 0 10-4 10-3 10-2 10-1 (a) (b) (c) (d) The constraint on N significantly alters the model behavior.

The system is under a high pressure of potential new species trying to migrate into it. N=const critical point The constraint on N significantly alters the model behavior. The system goes to an off-critical state as N decreases, preventing critical avalanches of extinctions. Extremal dynamics + Constraints -> SOC

Summary • We formulated and studied models that bridges the DG model and the BS model in order to identify a key factor for generating SOC phenomena in a biological evolution model. generalized model DG model link-based BS model • The applicability of BS model is questionable as the conservation of the system size is not satisfied in general.

Y.Murase & P.A.Rikvold New J. Phys. 20 083023(2018)